# nt.number theory – Is there any computational evidence of the analog \$ pq \$ of the Serre conjecture?

the $$pq$$ analogue to Serre's conjecture (see "Mod pq Galois representations and Serre & # 39; s Conjecture" – Khare, Kiming) says that if $$bar { rho} _1: G _ { mathbb {Q}} rightarrow text {GL} _2 ( mathbb {F} _p)$$ It is a mod $$p$$ Representation of Galois and $$bar { rho_2}: G _ { mathbb {Q}} rightarrow text {GL} _2 ( mathbb {F} _q)$$ It is a mod $$q$$ Representation of Galois, then yes $$bar { rho} _1$$ Y $$bar { rho} _2$$ they are odd, irreducible, branched into only a finite number of prime numbers and of the same "weight", then under some additional hypotheses, $$bar { rho} _1$$ Y $$bar { rho} _2$$ Both elevate to Galois representations coming from the same own form.

Before Serre made his conjecture (for example, a cousin), there was certainly a large amount of computational evidence, is there any computational evidence for this conjecture?

It seems that the easiest way to find examples of $$bar { rho} _1$$ Y $$bar { rho} _2$$ as described would be to take the mod $$p$$ and mod $$q$$ representations associated with a new form and one would have to build to find $$bar { rho} _1$$ Y $$bar { rho} _2$$ That do not arise in this way to find computational evidence.